Stable and Unstable Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation

Summary.

Investigated here are interesting aspects of the solitary-wave solutions of the generalized Regularized Long-Wave equation

$$u_t + u_x + \alpha \left( {u^p } \right)_x - \beta u_{xxt} = 0.$$

For p>5, the equation has both stable and unstable solitary-wave solutions, according to the theory of Souganidis and Strauss. Using a high-order accurate numerical scheme for the approximation of solutions of the equation, the dynamics of suitably perturbed solitary waves are examined. Among other conclusions, we find that unstable solitary waves may evolve into several, stable solitary waves and that positive initial data need not feature solitary waves at all in its long-time asymptotics.